Schatten Class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space $G\mathrm{/}H$ characterizing the Schatten-von Neumann classes $S_r(L^2(G\mathrm{/}H))$ for all $0<r\le \mathrm{\infty }\mathrm{.}$ We go on to provide a symbolic characterization for $r$-nuclear, $0<r\le 1,$ pseudo-differential operators on $L^p(G\mathrm{/}H)$-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space $G\mathrm{/}H\times \widehat{G\mathrm{/}H}\mathrm{.}$ Finally, we present applications of aforementioned results in the context of heat kernels.